George W. HartDepartment of Computer Science State University of New York at Stony Brook Stony Brook, NY 11794-4400 USA
The educational application of the 72-sided sphere shown in
Figure 2 is clear if one studies Euclid. It illustrates the construction
for a polyhedron inscribed in a given sphere (Book 12, Proposition
7) and approximates it arbitrarily closely, leading to the theorem
that the volume of a sphere is proportional to the cube of its
radius. Beyond this type of direct pedagogical use, these forms directly
present aspects of symmetry, balance and design that are central
to classical architecture. Pacioli discusses these issues in
Chapter 53 of his book and also mentions how polyhedral forms
can be used as a model for domes. In the twentieth century, polyhedral geometry has been found to be the basis for a wide range of designs, such as Fuller's geodesic domes, space structures, deployable buildings and many other types of "nonstandard architecture".[8] Today, progressive undergraduate architecture programs may include design labs that involve geometric model building, to develop students' understanding of space. However the wider culture is not very familiar with polyhedral models, because solid geometry is no longer featured in many high school curricula, and outside of school there are few opportunities to encounter polyhedra.[9] To increase the general awareness about geometry, it is valuable to build and display such models. Printed figures and computer animations can be widely reproduced, but three-dimensional models, when available, have much more impact because they are real and tangible. Pacioli records that he carried wooden models with him to use as illustrations when he lectured. We know that the value of such models was officially recognized in the Renaissance, because there is an entry in the accounts for the building of the Council Hall in Florence indicating that a set of Pacioli's models was purchased by the City of Florence for public display.
Both paper and wood can lead to attractive results, but paper is of limited durability and woodworking requires more time and skill. I wish now to recommend a third technique: 3D printing. This is a new technology in which a computer-controlled robotic device automatically constructs a physical three-dimensional object from the description given in a computer file. While relatively expensive at present, there are several companies competing to develop these methods, so the cost will lower significantly in the future. Currently, manufacturing industries and larger universities are beginning to use this equipment. In the future, I expect all universities and high schools will have 3D printing equipment. There are many educational applications for 3D printing, such as mathematical, chemical, anatomical, and architectural models. But it is especially inspiring that such a state-of-the-art technology may be applied to recreate and disseminate copies of historically important artifacts.[10] Three state-of-the-art 3D printing technologies are illustrated here. Each required several hours for a machine to construct, but only a few minutes of operator time to set up the job and start it. Of course there was a significant amount of work required for me to design the files that specify the forms, but that need not be repeated. As a computer scientist interested in making three-dimensional forms, it was natural that I wrote my own software for the purpose. However, many commercial 3D computer-aided-design tools would have been adequate for designing the files. The plastic model in Modern 3D printing methods go beyond anything even Leonardo
might have conceived, yet this technology can help his polyhedra
to remain educationally relevant. Students now can create and
display a mathematical "cabinet of curiosities", or
"instant museum" of these historically significant
forms. I can provide the computer files that describe their structure,
but it is an excellent pedagogical project to have students design
the files. Constructing polyhedra is a computer-aided-design
exercise that requires students to master precisely the sort
of geometrical knowledge that Pacioli was teaching 500 years
ago. However, instead of ruler and compass, the basic tools for
manipulating polyhedra on a computer are their
Euclid's
Kiss. For photos of other Leonardo-inspired models I have
made, and related geometric sculpture, see my webpage.
return to text2. Written in the
Italian vernacular, 3. Careful study
of the manuscript drawings shows that a few of the polyhedra
illustrated in Pacioli's treatise contain some geometrical inaccuracies.
Portions of the rear surface visible through the openings of
the front faces are occasionally misplaced. An analysis is beyond
the scope of this paper, and could only be very speculative,
but here are three possibilities one might consider: (a) Possibly
these were approached as character sketches, and not intended
to be as accurate as scientific illustrations; (b) possibly the
extant drawings are Pacioli's inaccurate copies of lost Leonardo
originals; and (c) possibly Leonardo did not have physical models
for some of the drawings. 4. One can not
help but be curious as to whether Pacioli and Leonardo had deeper
reasons for presenting some of the particular forms that they
chose—what might they have been models of? Several nonconvex
forms assembled from equilateral triangles, e.g., Figures 1,
3 and 4 were original to them, and their book does not explain
why they are being presented. Pacioli's text repeats the classical
associations given in Plato's 5. For a scholarly
study of the wide influence which Leonardo's polyhedra drawings
had and the connections between polyhedra and Leonardo's own
designs for churches and cathedrals, see Kim H. Veltman, with
Kenneth D. Keele, 6. Euclid, 7. Alan Tormey
and Judith Farr Tormey, "Renaissance Intarsia: The Art of
Geometry," 8. For an excellent
recent survey of how polyhedral forms have been applied in twentieth-century
architecture, see J. Francois Gabriel, ed., 9. For interesting
background on how models may be used in an architecture course,
see Pierangela Rinaldi, "The
Renaissance, Geometry and Architecture", 10. The technology
of 3D printing is advancing rapidly, so the best source of current
information is to do a search for "3d print" on the
Internet.
J. Francois Gabriel, ed., Plato, Euclid.
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